Monodromy of codimension 1 subfamilies of universal curves

Detalles del recurso

Descripción

Suppose that $g\ge3$ , that $n\ge0$ , and that $\ell\ge1$ . The main result is that if $E$ is a smooth variety that dominates a codimension $1$ subvariety $D$ of $\mathcal{M}_{g,n}[\ell]$ , the moduli space of $n$ -pointed, genus $g$ , smooth, projective curves with a level $\ell$ structure, then the closure of the image of the monodromy representation $\pi_{1}(E,e_{o})\to {\mathrm{Sp}}_{g}(\widehat{ \mathbb{Z}})$ has finite index in ${\mathrm{Sp}}_{g}(\widehat{ \mathbb{Z}})$ . A similar result is proved for codimension $1$ families of principally polarized abelian varieties.